[Paper Review] Sparse reduced-order modeling : Sensor-based dynamics to full-state estimation

Jean-Christophe Loiseau, Bernd R. Noack+1 more|HAL (Le Centre pour la Communication Scientifique Directe)|Jun 12, 2017

Model Reduction and Neural Networks14 citations

TL;DR

This paper proposes a sparse reduced-order modeling framework that transforms time-resolved sensor data into a predictive, interpretable dynamical system using sparse identification of nonlinear dynamics (SINDy). It achieves high accuracy in full-state estimation by lifting sensor features to a dynamic space, identifying sparse nonlinear models, and using optional PIV snapshots to build a highly accurate local linear mapping to velocity fields—outperforming traditional POD-Galerkin models in transient cylinder wake simulations.

ABSTRACT

We propose a general dynamic reduced-order modeling framework for typical experimental data: time-resolved sensor data and optional non-time-resolved PIV snapshots. This framework contains four steps. First, the sensor signals are lifted to a dynamic feature space. Second, we identify a sparse human-interpretable nonlinear dynamical system for the feature state based on the sparse identification of nonlinear dynamics (SINDy). Third, if PIV snapshots are available, a local linear mapping from the feature state to velocity fields is shown to be orders of magnitudes more accurate than optimal modal expansions of the same order. Fourth, a generalized feature-based modal decomposition identifies coherent structures that are most dynamically correlated with the linear and nonlinear interaction terms in the sparse model, adding interpretability. Steps 1 and 2 define a black-box model. Optional steps 3 and 4 lift the black-box dynamics to a 'gray-box' model of the coherent structures, if non-time-resolved full-state data is available. This gray-box modeling strategy is successfully applied to the transient and post-transient laminar cylinder wake, and compares favorably with a POD model. We foresee numerous applications of this highly flexible modeling strategy, including estimation, prediction and control. Moreover, the feature space may be based on intrinsic coordinates, which are unaffected by a key challenge of modal expansion: the slow change of low-dimensional coherent structures with changing geometry and varying parameters.

Motivation & Objective

• Address the challenge of creating accurate, interpretable reduced-order models from limited experimental sensor data without full-state measurements.
• Overcome limitations of traditional POD-Galerkin models in handling transients, parameter variations, and slow structural deformations.
• Develop a flexible, data-driven framework that transitions from a black-box sensor model to a gray-box model of coherent structures when full-state PIV data is available.
• Enable robust, sparse, and physically interpretable dynamical models for estimation, prediction, and control in fluid dynamics.

Proposed method

• Lift time-resolved sensor signals into a dynamic feature space using user-defined or intrinsic coordinates to enhance model expressiveness.
• Apply the Sparse Identification of Nonlinear Dynamics (SINDy) algorithm to identify a sparse, nonlinear dynamical system governing the feature state, ensuring interpretability and low dimensionality.
• Use non-time-resolved PIV snapshots to construct a local linear mapping from the feature state to full velocity fields, achieving orders-of-magnitude higher accuracy than standard POD expansions.
• Perform a generalized feature-based modal decomposition to identify coherent structures most dynamically correlated with linear and nonlinear terms in the SINDy model, enhancing physical interpretability.
• Integrate physical constraints (e.g., energy preservation) into the SINDy identification process to improve model stability and performance.
• Apply the framework to the transient and post-transient laminar cylinder wake at Re=100, validating performance against POD-Galerkin models.

Experimental results

Research questions

• RQ1 Can a sparse, interpretable nonlinear dynamical system be identified directly from time-resolved sensor data alone?
• RQ2 How can non-time-resolved PIV snapshots be leveraged to improve the accuracy of full-state reconstruction from a low-dimensional sensor-based model?
• RQ3 To what extent does the proposed framework outperform standard POD-Galerkin models in capturing transient and post-transient dynamics?
• RQ4 Can the framework be extended to reveal physically meaningful coherent structures through a feature-based modal decomposition?
• RQ5 How does incorporating physical constraints into the SINDy identification process affect model accuracy and stability?

Key findings

• The SINDy-based sensor model accurately captures the transient dynamics of the laminar cylinder wake, with predictions closely matching direct numerical simulations.
• The inclusion of PIV snapshots enables a local linear mapping from feature states to velocity fields that is orders of magnitude more accurate than equivalent-order POD expansions.
• A constrained cubic SINDy model, informed by physical considerations, outperforms both unconstrained and lower-order models, achieving the best fit according to the AICc criterion.
• The generalized feature-based modal decomposition successfully identifies coherent structures most correlated with the linear and nonlinear terms in the sparse model, enhancing interpretability.
• The framework demonstrates robustness to noisy data and can be applied to systems with varying geometries and parameters, as the feature space is invariant to slow structural changes.
• The discrete-time SINDy formulation successfully identifies a stable, first-order sparse vector autoregressive model that accurately predicts the phase-space trajectory of the cylinder wake.